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Let $D \subset \mathbb{C}^n$ be a bounded symmetric domain. It is known that $D$ can be realized as the unit ball of some complex norm $||\cdot||$. Using the Bergman metric on $D$, one can define a triple product on $\mathbb{C}^n$ which enjoys many nice properties. In the paper, a precise formula for an automorphism of such a domain is stated in terms of this triple product (equation 1.8). The reference given in the paper deals with bounded symmetric domains in Banach spaces and involves a lot of machinery. Does anyone know any reference where the automorphisms are described in the finite-dimensional case with proof? I have already had a look at the standard reference, but it does not seem to have it.

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In S. Helgason's oldest large book, "Differential geometry and symmetric spaces" (?) from the 1960s, symmetric spaces and bounded symmetric domains are discussed in detail. Also, I. Satake's book from c. 1980, I think in a style of the sort you're wanting.

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